HhttHtHCEIC/94/204

IWFCRNATIONAL CENTRE FORTHEORETICAL PHYSICS

QUANTUM SYMPLECTIC GEOMETRY1. The matrix Hamiltonian formalism

A.E.F. Djemai

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

MIRAMARETRIESTE

Mil: mm. m <

IC/94/204

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUANTUM SYMPLECTIC GEOMETRY1. The matrix Hamiltonian formalism

A.E.F. Djemai 'International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

The main purpose of this work is to describe the quantum analogue of the usual clas-sical symplectic geometry and then to formulate the quantum mechanics as a (quantum)non-commutative symplectic geometry. In this first part, we define the quantum sym-plectic structure in the context of the matrix differential geometry by using the discreteWeyl-Schwinger realization of the Heisenberg group. We also discuss the continuous limitand give an expression of the quantum structure constants.

MIRAMARE - TRIESTE

July 1994

'Permanent address: Laboratoire de Physique Theorique, Universite d'Ora^Es-senia,31100, Oran, Algeria.

CONTENTS:

t. Introduction and motivation

2. Classical symploctic formalism ID

3. Weyl-Wlgner-Moyai formalism 12

4. Wayi-Schwingar realization of the Heisanberg grnup

5. Matrix formulation of a quantum bosnnlc oscillator 42

6. Matrix differential goomstry Si

7. Quantum £matrbO symplactlc formalism 60

7.1. First approach 60

7.2. Second approach 63

8. Discussion and per^sctlves 76

9. Rsfsr 77

S1 '

i . Introduction and motivation

It Is VJBII established new that it is passible to give a complete descrip-

tion of Harnlltanian mechanics In the context of Paisson symplectlc geometry

[ i j . On thQ other hand, slncg the appearance of quantum mechanics at now

several attempts were presented to give a precis© interpretation af the <pxm-

tlzatlan phenemenon. This latter refers substancially to the construction of

a quantum system which admits the classical one as Its limit when the

Planck's constant ft tends to zero.

At the beglnlng, quantum mechanics was Interpreted as a statistical

theory over phase space. Founding upon the Wayl quantization procedure [2],

Wlgnsr gave an Bxprassion for a phase space distribution function [3}.

Most lntargstlng developments of the Weyl-Wigner approach are due to

Moyal [4], wha Introduced the sino-Potssan brackQt [ or Moyal brackst ), for

Functions Dn phase spacB that firstly corraponds to the commutator of

quantum mechanical operators associated to these functions, and secondly

goes into ths usual Poisson bracket at the classical Elm i t : ft • 0.

In other repects, the deformation thgary of algebraic structurGs began to

be studied In the fifties [5]. In the seventies, it was made usa of this mathe-

matical machinery In the context of symplectlc geometry [6], and In quanti-

zation [7] where it was realized that the Weyl-WignsHvioyal quantization

procedure can be fitted Into this theory as a very particular case. Lately, we

assist to a revival of this technique of quantization by deformation [S].

In ths context of C -algebras, ths set C Ovfl of smooth functions on the

classical phasa space (CPS) M t n Df some classical dynamical system forms

a commutative associative Poisson-Lie C -algebra equlppod with a polntwise

product " . " and with a Folsson bracket " { , } ". The undarlylng differential

geometry is endowed with a {classical} commutative sympiecttc structwe.

So, quantization of a classical dynamical system appears In this framework

as a breakdown of ths commutatlvlty symmetry Df the classical C -aJgsbra.

In fact, the quantization by deformation procedure consists to maintain the

structure of the CPS and to change the algebraic structurs defined on it by

deforming the polntwlse product and the Polsson bracket Into a star-product

11 ^ " and a Mayal bracket " { , }fi " respectively, making use of the Weyl

correpondeocs.

In other hand, ths non-commutative tort [9] may ba considered as defor-

mation quantizations of ordinary tori for an appropriate Poisson structure

[10], It -vas shown that ths action ( by translations ) of an ordinary torus

T^ - ??f2? an the non-commutative torus A - (f^cfy makes this latter

a non-commutative symplectlc manifold with a smooth dlfferentlable struc-

ture Dn which Carinas [11] showed how to extend the apparatus of the usual

rsxsnrw

differential geometry Involving comsctlon, curvature, Cbern classes etc... .

For Instants, the Helsenberg group may be represented as an extension of

tha abBlJan doubig cyclic group Z^JSZ^ and than, finite representations of

this group which Is realized by two eJemnts U and V satisfying :

V.U = w U.V

where u = exp[127r/N] £ C, are obtained by considering U and V as NxN-

mstrJces such that :

(th9 sot ^ = 2 Cm ^ ' . V ^ [ =: J 2 <L T ^ L with

—»> *m = (mitma) £ Z /[Q,Q), C^^ rgpresgntlng complex coefficients In the basis

m{ T_^) and :

m

T_,=4F-<^ * u .v .

will descrlbB a non-comrnutatSve Cor quantum } two-tarns. In fact, if we

Introdt^e a Lie algebra structure by means of the standard commutator, we

obtain the so-called class of trigonometric slne-Lle algQbras (or tha quantum

Tori Lie algebras) [12] with :

IT* »Tm n

where ths two-cocycle 33 defined such that :

Jj { m , n ) = - a,( n , in } = - ^ m x n = - ^ j -

corroponds In the JanguagG of [11] to an siemant of ths first cyclic cohomo-

logy group and wll] sgrva tD deflna ivhat we wiJl call sJmpJJcJa^ symplectlc

structiro on this class Df non-commutatlvB algebras.

The N —• » limit A^ is similarly defined by:

m n m + n

More to the point, there exists a central gxtanslon of this class of quan-

tized by deformation Polsson-Us algebras on the 2-torus [13]:

» 4 sin[2^f ^ , ff}] T_^ ^ + a*. Sftf cmn m + n m + n , o

[ T_^ t c ] = [ e t c ] = 0m

with a = [ai.Si) 6 C and c being the central element.

Therefore, the starting point of the non-commutative dlffgrGntial {symp-

Isctic) gpometry consists to replace the abstract commutative C -algebra

CftM) af smooth functions an a commutative [symplsctlc) manifold M by

a non-commut&lve C -algebra A of functions an a non-commutative [symp-

isctic) spaces. It's clear that, in the non-commutative sympJectic case, one

must define ths non-carrrmitativg (or quantum) analog of the commutative

(or classical) symplectlc structure sunh that ana masts this latter From the

first by means of a commutative limit.

In addition to the Connes' approach [1 i ] , there exists another on© due to

Dubois-Vlolette & al [14] which dlffarg From tha first one essentially in the

definition of the noncvmmutatlve generalization Q of the differential algebra

of differential forms. In this approach, s nan-commutative symplsctlc struo-

2

tura for A is an element <*> of Qrvr (A),where QrwLA) is the smallest diffe-

rential sub-algebra of the complex C{Der'A);A) and Der(A] the Lie algebra

of ail derivations of A. The element w must satisfy the following conditions:

*-For a given H € A, there Is a unique derivation hann(H] 6 Der(A) such that:

U X , ham(H) ) = X[H) , for any X 6 Der(A);

**- u> is closed.

It's easy to see that the existence of the commutative limit is ensured.

In this context, It v ss shown that the Heisenberg algebra A. defined asit

a C -algebra with unit and generated by two hermitlan elements p and q :

will cx>rrosponds to ths t-^o-dlmerslonal quantum phase space CQPS) with

a non-commutative sympleotic structure given by : [15]

u = ^ — [....[dp,p] pl*E—.[dq»q]*....,q]n Uft) (n+l)l n n

'which reduces in tha cammiAattve limit to tha ordinary one : dp «dq. This

shews that ^unrtum mechanics can be understood as a n-sn-ci

symploctlc gaametry.

Let us finaiiy remark that there is a deep link between the theory of

quantum grcups and the star-deformation. Effectively, Dubols-Vlolette

showed in (16] that the C -Hopf algebra af representative elements corres-

pcndlng tD a compact matrix quantum group 117] Is Isomorphlc, as C -

co-algebra, to the C -algebra of representative functions on the correspon-

ding classical compact group. This isomorphism appears as a generalization

of the Weyl correspondence [2].

The purpose of this work Is precisely to Investigate this question in

some details. The paper is organized as foliovvs. In section 2, we recall

•rtafiy some notions of classical syrnplectin geomBtry. In section 3, we

revlaw tha Weyl-Wigner-JvfoyaJ formalism In section 4, we present tha Weyl-

Sehwlnger realization of the HBlsenberg group as an extension of the abeilan

double cyclic group Z^xZ^ and discuss the Importance of the choice of the

Schvlngsr basis. At this level, it appears that oriB may study Physical

situations with various number of degrees of freedom basing upon the prime

decomposition of the N, In this context, we study the case of d degrees of

Freedom and examinat© tha continuous situation, where it appears that the

Schwlngsr basis leads naturally In this limit ta the generators of the von-

b

ccnxratativs Halssnterg algebra. This will parmJt to define the trua qtsntum

sympiectic structure on tho quantum mechanical Hllfaart spaca. Considering

ths Schwlngsr basis as s Fourier basis, one may easily see that quantization

Is deeply tied to tha Fourier analysis. In this ccntsxt, ws construct the

discrete version of the Wsyl-WlgnerMoyal formalism and present expildtely

thg casa of N ~ 2.

[n section 5S v/s treat the matrix Formulation of the 1 dimensional

;qiantum bosoolc oscillator and give tha axtonsion to d degrees of freedom.

WQ also romark that the spectrum of the Hamlitonian matrix prasants an

Intsrostlng faature dua to the sppgaranca of a twofold degeneracy In tha cas©

N Is avon.

apparatus of the noncommutativsr diffsrential geomatry of matrix

algebras Is presented In tbs section 6. The matrix aigsbras M^(C) are ths

most slmpig non-trlvoai example given by DubolsVioiette S al in their

approach to non-commutatlva diffarantJal geomgtry, [sea [30]). We also

present the GBSB of N = 2. In section 7, we dlsciss the different matrix

approaches to describe a qiantum sympiectlc formalism. Firstly, we review

the DubolsVSolette sppraach. Then, wa adapt the Wgyi-Schwlngor realization

of the Helsenberg group to this formalism and discuss ths continuous limit.

The section 8, Is devoted to some concluding remarks and porspectlvas.

2. Classical syrnplecUc Formalism : [1]

Let us consider a dynamical systam evoJving in a classical phasg space

M 2 d = T^ffi") with local coordinates e3 = fq^p.), a = l,...,2d, I,j = i,...,d,

in such a way that they obey the fundamental Poisson brackets :

' j } p j ( i ) >

is compactly summarizod ss :

{€ , e } = w C2),

vvfsers the antisymmetric matrix wa given by :

P),

Is ths invgrsB of the symplactlc matrJx <*)•, l.a. :

"ab-^C W-

In fact, wg. represents the componQnts of thg clasGd non-dogenerate

symplectlc 2-Form u :

tj = i ^ a b dca Adgb = d6 = dpx «dq1 (5),

with 6 = pjdq1 is the canonical {Llouvllla) i-form. This defines the Hamil-

Ionian structure on the phas? space M.

Indeed, Far any smooth function H(q,p) = H(g) DPI M one associates the

Hamiltonlan vector field :

whsrs 3^ = 3/3 g". Tha components X^ that are given by

(6),

XHb=(3aH(c)).<jab C7),

permit to deduce the Hamlltonlarf s canonical oquations :

€S = XH&(€) (3).

It Is easy ta see that ths Polsson bracket involving two arbitrary

functions F(e) and GU) on M is given by :

{ F(€> , GCe) } p = - o>£ XF , ^ ) = w[ X Q , X F ) = Xp(G) = -

Henc», th© sat (^(M) of classical observabtes possesses thg structure of

a Palsson-Lle algebra Ao = ( C°°[M) , "." , U p ) equipped with two Internal

laws, ths polntwisQ product "." and Polsson bracket {,}p, in such a way that

tha Jacofci Identity :

{ { F , G } p , h } p + { { H , F } p , G ^ + { { G , H } p , F } p = 0 [10)

is equivalsnt to the relation :

and the Lslbnltz rule :

{ F , G.H } p = [ F , G } p . H + G . { F , H } p (12)

guarantaas that ths Hamiltonlan vector field is a derivation :

XF[G.H) = XF(G] . H + G . XF(H} (13}.

3. Weyi-Wignor-Moyal formalism :

Let B = Ru be a configuration space on which ans-partlcie system moves

and 1st M = T (B) ^ R 4 be its associated classical phase space (CPS)

with local cooniinatQs (q ,p.) , X,J = i,...,d. The CPS Is then equipped with

a Llouvlile i-form 6 - p.dq and a classical symplectic structure defined by

the closed non-dagsnarate 2-form <*> - dp, * dq .

The Weyl map consists ta associate to a classical cbssrvabla F{ q , p )

€ AJ = ( C^fM) ,"." , U p ) s quanium observable O^i q , p*) acting as an

2

operator on the quantum mochanlcal Hilbart spaco L fB), by means of the

following operator Fourisr transform , [2] :

£2TT)' »4d [14)

where p and q are the self-ad Joint operators on L (B) obtained by the corres-

pondence principle from the classical variables p and q. Tha operators p and

q generate the nan-commutative Fundamertal Hotsmhsrg algebra :

i q1 , ^ 1 = i PX ' Pj 1 = D . [ q1 , Pj J = ift ^ il [15)

which is the quantum version of the algebra (1). In fact, the quantum mecha-

nical Hllbert space represents the so-cailsd quantum phasQ space (OPS).

Jistly, the aim of this paper Is to try to define tha quantum analog of the

usual classical symplectic structure.

13

The Wgyj msp :

F[ t, ?) > Op( q , p ] (16)

is lnvertlbls, i.e. thsre is a i to 1 correpondence between functions on CPS

and their analogs on GPS. In Fact, the functions F( q t p*) that are often

called Wigner fisictions are defined as ordinary Fourier transforms of the

so-calJed Wigner fcnsltlos Dp[ a*, &*), [3] :

F( q , p ] = — ~ r - da d& [>p[ a , 6 ] e L H 4 J (17).

The Wigner Functions F ,which sra o-numbsrs, are also ca]3od symbols of

tho assoclatgd operators Op, [IB]. IF F ( respectively F ) denotas the usual

Fourier transform ( respectively the operator Fourier transform ), we get :

F = F[ Dp ] = F [ T 4 [ O p ] ] (1B.21.

The densities Dp includg commonly Dlrac deltas or / and their deriva-

tives. In the sense of [3], to any quantum mechanical statB vector |^>, which

constitutes a 1-dimensional sub-space of OPS, we may associate a Wigner

Function F( ^ , p*} daflnod as a symbol of the projector operator !^><\J/i.

Then, the corresponding Wigner density becomes a psoudo-probabllity distri-

bution (In ganaral, not positive definite) which reproduces the classical pro-

bability density v»hen h • 0.

ThG domain of the operator Fourier transform F has beer, extensively

studied in the literature, [19]. That Is for the Weyl-Wigner transformation.

Now, we will discuss the developments of this approach due to Moyal [4].

It is well known that the differential geometric structure of a manifold M is

perfactly determined by the propert'gs of the algebra Aa = C^fM) of smooth

functions DP M. Hence, to describe correctly thB classical Hamlltonlan

mschanlcs ons must study the symplecttc dUforertlal geumotry of tho CPS,

or equlvaJently, ths Folsson-LlG algafara Act C°°[M) , "." , [,}p ) of commuter

tlve classical observables on M.

Now, to describe quantum mochmtcs it Is straightforward to think of a

nan-commutative generalization of ths above geometry. This idea was

percelvBd at the advent of quantum mechanics [20], where this Jatter appea-

red as included In the framework of a non-commiAativQ varslon of the notion

of Polsson manifold [21], which will represent the QPS.

In any c&S6,the algebra A of quantum observables should be equipped with

VMJ Internal laws that can be compared with "." and [,)p in A,. String that,

in the statistical Wsyl-Wlgner formulation of quantum mechanics one don't

manipulate operators but their symbols* ons may think of a twlstod version

of the two internal operations of AB. Rsmark that ons of the main features of

this approach is that the framework In which one describes a quantum

. *-». ;^iaiii'::«*> fa- •>• -

system Is the same ss the ano usad tD describe tha corresponding classical

system, i.e. the sympisctic manifold (M,<:J). The only important difference Is

the chDlcs of algebra Df functions on M, namely th© choice of Internal laws

that one must equip this algebra. The most simple choice consists to deform

tha commutative product ".M and the Polsson bracket {Jp Into a non-commuta-

tive product!l " " and a twisted Polsson bracket {,} respectively,by means of

a deformation paramstsr JJ such that, for a particular value y0 of ut ore hss:

llm F ' v G = F.G (19.1),

v — • v0

In GUT contaxt, the deformation paramstBr Is not slsa than th© Planck's

constant h and the classical limit that guarantaes the passage from th©

twisted algebra Afi[ C*tM} , " xfi" , {,}fi ) to the classical one Ao Is h — • Q.

Then, a quantization may be viewed as 3 ons-parameter deformation of

the associative pointwlse product "." in the direction of Cns Polssan bracket

fn such a way th&, for any two olomorts F and G of some sultabte sub-

algBbra So of Ao that Is stable under "." and Q p (Far Instance: functions of

compact support, Schware functions, polynomials, etc...), ©qu.'s {19} hold.

In fact, this quantization is not unique. There are classes of quantization

such that to a classics) algebra Ao DOG may associate a Family Df quartum

algabras A^, [22].

The first example of star-product was given by- Moyai [41. Starting from

the WeyJ map (16), consider the product af twa cparators Op( q , p ] and

OQ{ q , p ) and try to define the resulting (deformed) product af the

associated symbols F and G. Then, from:

OG(q, jf] =O^[qu, ?) f2D]

[21)

and using CIS) and the Glauber formula :

A+B= B

we get in terms of Wigner densities the following formula

R2dX 9

a ] / 2 (22).

Knowing that the product " ." is tied to the convolution product" >:" by :

F [ F * G ] = F [ F ] . F [ G ] [23.1)

or,

where,

F .G =

=— -

[ G

dy F[y).G[x-y)

[23.2),

(23.3),

than, it Is possible to identify equ. (22) with a twisted convolution product :

DH = ° F xft DG ( 2 4 >'

From equ. (24) ,Dn9 naturally deduces a relation between the star-product

11 *fi " and the twisted corwlutlon product" xft " analog to (23.2):

H = F ' G = F[ F' {¥} xr F [ G ] 1 (25).

The Moyol product is s star-product ^ defined on the space (f^iMjt] of

forma] series in h with coefficients in CW(M, such that :

5L -^ (26),

where 9a = 3/3ta , £3 = (q1^,) with a = l,..,,2d, i,j = l,...,d, wab Is the

Inverse of the symplectlc matrix (see (3)) and P Is an operator defined fay:

P°(F,G)=F.G [27.1),

?(F,G} = [ F , G } p (27.2),

T \ nn n [ 3 . . . 3 h G) C27.3),

, 0 = 0 for n > 0 if F or G Is a constant in C^tM.fi), [27.4),

2 Pn(PmCF fG),H)= 2 P°t F , P ^ H ) ) [27.5).n+rn=t n+m=t

The equ. (27.5} gxprsssgs the associativity af the non-commutative

algabra C^(Mtfi]. Moreover, It's easy to see that the Mays] product defined

by equ. (26) satisfy ths axiom (19.1). In fact,by means of the precise choice

(27.3) Df suitable forms of the bidifFerentisI operators :

P" : C tM.ft) x CffHfi) —*Ccc{M,fc)

that satisfy (27.1,2,4 and 5), we have completely defined the Moyal product.

Now, to emphasize the Polsson-Lis algebra stnjctura of ( C°°[M,ft) , "^ ),

one mist define s deformed Polsson brocket { , } , obeying the axiom (19.2).

IT© Moyal brocket is deduced from the commutator of two Dparators :

[ Op , OQ ][ q , p ) =: lft O{ F ^ G j £ if, p) = lft 0^1 J , ? ) [2SJ.

Then,

H(€) =: { F[e) , Gfe) }ft =:

(-1)n

2

slnt - 2 - P(F,G) )

2n+l

F(e). sln( . G(G) £29).

It*s cJsar that the axiom (19.2) Is FuJfllied. Moreover, this dsformod

bracket obeys the Jacob! Identity :

{ { F , G Jft , H Jft + { [ H , F } h , G }fi + { { G , H }ft , F ) h = 0 £30).

It also defines a derivation of the Polsson-Lie algebra ( C*(M,ft),"A,[,fy. )

with respects to \ , l.e obeys tD the Leibnitz rule :

[ F , G \ H } h = [ F , G }ft *fl H + G «ft { F , H }fi (31).

FinalJy, one has the following correpondonces :

operator classical star-deformsd

0 4c&xjnlcal quantization

1ift"

[ . ] • •

WByl-Wlgner-Moyalquantization

4. Weyl-Sdwingar realization of tha Heisonberg group :

If V/Q represent tha Hslsenberg group ss an extension of the absiian

doubia cyclic group IZ^SZLj than, finite representations of this group v^ich

is realized by two unitary operators U and V satisfying the basic relation :

V.U = u U.V [32),

with CJ a complex number, are obtained by taking for U and V NxN-matricss

(We use the sama symbol for tha operator and its associated matrix) such

that :

= V " (33).

By taking determinants in (32), Jt follows that:

o = axp[-jj- [2¥fi/N)]=exp[i27r/N] (34).

For each integer number M, thsrs is one realization of the Hbisenberg

group. Let { |or. > , k € Z } be s basis of orthanormalized kets for tha space

of states. Assuma that the ia^> are sigenksts of the operator V :

V|Lrk> = vkitrk> / vk = wk (35.1),

V1" jak> = wk m |ffk> = * V™ - 5 d™ lak><tikl C35.2).

Than, U is defined by :

U lak> = itfk+t> C35.3),

with,

!crk+N> 3 l-arfc> (35.5).

Those definitions ensure that U and V satisfy (33). In this basts, the

matrices V and U are given by :

V =m

0

Ql

• , U =

(?)•

•

•

D

i

1 0..•

m

0

01

• . o

• • : »

D

[35.6).

Choosing, far the space of states, another basis { 1^) , k £ Z } formed

by the eigonkets of U, one has :

/ 4, = .^k (36.1),

and,

with,

C36.2).

[36.3),

[36.4),

(36.5).

In this basis, the expressions of the matrices U and V arg inverted in

(35.5). In any case, art lndgpendgntly of the chalos of the basis, U and V

obey (33) and -verify :

^ ™ m n P " (37).

r » .»r « «••-

It is easy to SBB that the integers k,m,n are defined modulo N,

Any element W of ths operator* algebra of the Hsisenberg group can b©

determined, up to a scalar factor, by a triple DF intagBrs [m,n,p) such that ;

W = U m . V n u p £38).

These monomials in U and V constitute a complete basis for all quantum

operators related to soma chosen PhysIcaJ system. The abova expressions

are Invariant undBr the simultaneous transformations :

U — > V , V — t l S * and m — m , n > - m [39].

So, let us dBnote by T the operators that are Invariant under this

symmetry :

Tmn = J ™ / 2 L/^.V" (40).

These elements, which satisfy :

n\ _ t / 1 np _ fjTn nn _ ji rr>~ = nn+ _ npon * mo * oo * mn mn -m,-n

form a complete orthDnorma) basis of the group algebra with the properties :

(42.2),

tu^foktefrsss : T ^ = H)H 1 H2.3),

In tha basis { \a£ , k 6 Z ^ } where V is diagonal, ths apgrators Tm n

tha Form :

mn' V , .n£2k+m)/2 (43),

and iU corrosponding matrix NxN is given fcy :

mn

0.0 0

mx(N-m) 0

1 0 0.D i/1 0.

0 0 ,2n

0

0

.0

0. •fln{N-l)

°{N-m)xm

a

the aparators U and V have baan Jntroducod by Weyl [2], it

Scfiwlngor [23] who described quantum machanics in this formalism.

Any operator A belonging to tho group ajgebra will be written In the

Schwlngsr's basis { T ; m,n = O,....,N-1 } as :

A =m.n mn

(45),

with cDrnplex coefficients :

3 " l r L 1 mn'AJ " ' ! m n ' A J (46),

and with,

(47.1),

23

Tr[A] = a00 = A°° (47.2),

Tr[AtA]= 2 la171"!1 W7.3).

Here, the trace defines Gn thB operator algebra sn intamai product givon

b y ;

< A,B>=Tr[B^A] [48),

and consequently It follows the metric :

In this metric, the complete symmetrized basis [ T ) Is orthonormal.

Thon, the sst of operators Forms & metric algebra, with unity.

From now, we will use a more compact notation such that for m^m^

j...€ Z, one has :

m = (mt)mj) , n = (n,,na) , o = (o,o)

-> _ • -*-> C50-1).m + n — (rn^+n^mj+n^ , m.n —

and,

m x n =: r a ^ - mjOi (50.2).

Then,

—T— m,i m3

f_^=u>" U . V [51.1),m

_• = T+_+ = T _> [51.2),m m -m

_ __ .A] ffil.3),m in

E , ,= -^ Tr[T^.T_>] = J ^ €51.-4),rn n m n m n

^ f ^WV mn

For aach 'value af N, the operators T_^ realize a raprasentation thatm

makes use of a two-cacyc]9 a2. OccurJng frequont]y in quantum mechanics,

thBSB represGntations are callod proftettvs representations and in v,hlnh, the

operators T_^ dbey a generalizod composition law : [24]m

_ _^ ^ ^ ] T_^ _ (52),

n m m n m-f-n

with (ses C50.2)),

£ia{ m , n ) = -CT- m x n = - aa( n , m } [53).

m4 nt pi m3 n2 p z

In general, any two elements W, = U .V CJ and W3 = U .V u

will satisfy the following Helsenberg group defining relations In tsrms of

tripies :

= (m^m, , ni+n3 , ps+pj-f-J^na-manj)) (54).

From the associativity property [42.1), W9 gat the following consistency

condition :

,p j + cfjijn ,n+p ! - iT3iTTi ,n ) - U irr.QuiZ)) \uu).

Quantum mechanics also makes iiss of the one-cochain, [24], Indeed, the

action of the operators T . on a state iotr > is givsn by :m K

ml , ^ ( 5 6 ) >

with,

3(2k+m,) (57).

It turns that the so-called fundamental cocycle a3 is givon by :

az= Aoti - ai{k+n,;m } - ar rKjm+n ) + ati(k;n ) (58),

'whore A Is a nllpotont derivative fcobouidary operator] of some cohomology

giving information on the projoctlvs representation under consldaratlon, [25].

In fact, It has been shown In [25] that cr4 and as glvan by (5?) and (53)

respectively result from the action of algebraic cachaJns on the group

elements :

k ; m) = at( k ; T J (59.1),

a2l nf, n) = ati k ; T_^, T_ ) (59.2).m n

The nllpotent derivative A Is doflned on such cochalns. Thts, if a4 Is exact,

i.e. or! = Aot0 with a0 some Q-cochaln,o:t may bQ aliminated by adding a phase

cr0 to the wavefunctlone. When <ij is exact, It may bB eliminated by redefining

the operators In such a v/ay that thay appsar stats-Independent. It turns

actually that a3 = Aor4 (see (58)),so that Aas = 0 [see (55] j . This means that

cr2 Is a 2-cocycle callad the fundamontai cocycls. It's clear from (531 and

[57) that CT4 depends effectively on the state label k while ax does not.

Moreover, ths Fundamental cocycle will define a simpltclat symploctlc

structure on the lattico quantum phase space (LQPS) analog to the classical

one on tho usual CPS, while the i-ccchain ax will play on LQPS the role of

the Liouvllle canonical form 0 on CPS. In fact, LOPS is the lattico torus

described by tho double integer index m = fm^mj] that label the operators

belonging to the SchwlngBr's basis { T_^}, and in which, the area of each

melementary lattice Is equal to 2tf/N tending to zero when N —* » .

Therefore, the translations Issued From the action of the operator T fc

m

on LQPS may be Interpreted as unity multiple elementary gaps In the two

basic directions (U>V).

The choice of the symmetrized Schwlnger basis f T_^} for the Weyl rea-m

Uzatlon of the Heisenberg group is based essentially on Its particular Inte-

rest due to Its many remarkable properties, such that:

Firstly, for N=2, the T_^s reduce tD the Paull matrices [seer«34)) :m

m _ 11 u _ fl _ „ n p _ f OI 00 ~

C i l

27

fa fT « = l = < r 2 , T M = j = < 7 3 [60 ] .

U 0J [0 -l.J

For N ^ 2,tha Schwinger basis Is a preferable basis admitting additive quan-

tum numbers.lt provides the Finest grading of the Lis algebra gl[N,C) [26].

Secondly, there are many helpful relations, such as:

2 T^.A.T _^= [Tr[A]} t {61.1),r\ mm

, .Mm-n] _ rfnn(J -0

TTilrdly, because of the two-foldsdngss prapgrty (42.3), the T_^s c»ns-m

tltute a double covering mad(N) of the torus, with [rn^mj) playing the role

of coordinates mod(N). U and V may also be seen ss global coor-

dinates with values in ZKT&Z*, appearing as non-commutative point functions

because of the protective character of the representation, It was shown in

[25] haw closed paths on LQPS iaad tD open paths on operator space and

how this fact is related to non-cammubattvity;

Fourthly* one may study Physical situations with various number of degrees

of freedom following the vaius of N. For instance, Whan N is prime number

the pair (U,V) describes one degree of freedom taking on N passible values.

If N is not prime, then It is a product of ,say, d prime numbers :

N = N 4 x . . . x N . x . . . x N d (62),

and the Schwingsr basis becomes a product of d independent sub-basJs, one

for each prima Factor, that Is, ana for Bach dagrsa of Freedom :

8 { m nj} ®_ [J J «J

m,nv2 m4 rrj J U J.V ] (63),

where m . l n J = O , l , . . . , ( N , -1) and,

ut = sxp[ K27T/N.] ] (64).

This result Implies a classification of the quantum degrees of freedom in

tarms of prime decomposition of N. If ws want to work with two or more

degrees of freedom, one must use for N a well-chosen non-prime valuB.

Fifthly, the usual situation of the position q = ( q , , , . ^ ) and tha momentum

p = (p4t...,Pj} operators Is recovered if we CHDOSB :

" j 4 •^HnJ = V = BXPf i(27z/N.)* n^.p,] £65.1),

J J

Tm Q = U -J = sxp[ -fc (23r/N,J*m..qJ] (65.2),

j = l,....,d. Then, equ. (63) becomes :

J (66.1),

we have used the following compact notation ;

z dn )

(66.2),

—+ -+ ^ z dwJth m = [ m1( rn2 , m^ ) and n = [ n , n , . . . . , n ).

To pass to the continuous casB,it Is sufficient to take to infinity the torus

radii, while N • eo if d = 1, or N. • <x> with j = i , . . . ,d when N i s not

a prime numbor [d > i ] . In this limit, one has :

* nJ • a J (57.1),N — • »

so that :

and,

4 (27T/N.]1 m , • - b , (67.2),

N jnaxpt-SU-J.b.] (57.3),

U J • U[ b. } = sxp[ -1 b j ] (67.4),

J 1 1V •Vta-J)=(wp[iaJ.p,] [67.5),

N. —• (» 3

NJ

[67.6),

where a = (a ,...,a-\...,a 1 and b = (fcu b4,...,bj ara sams constant dual

vectors characterizing translations :

q >q*+a* (68.1),

p* • p*+ 6* (68.2),

affected by tho aporstor T(s ,& } on thG 2d-<Hmgnslonsl phase spacB Vt/hich

Is a Hllbert spaca corresponding to the true quantum phase spaco (OPS).

To obtain the aqu. (67,6], ore has simply used the daflnlng commutation

relations of the Heisenberg algebra (15) and tho Glauber formula (21).

The choice of tha Scfwlngsr basis (66. U leads then naturally under

a continuous limit to tha generators T(a\1?) Df thg non-QommutatlvB Helsen-

borg algabra obeying a generaJlzgd DDrnpositlon law analog tc [52) :

$) = exp{ Im2[$$);(£>£)} } T(t+c >t+£) {69}

T[ a*, t) ; T[

a*, &*) x ( ?, d*) = \ {t£- tc) (70).

Here, one must do the Following rsmark. The 2-cocycie a:3 given by equ.

(53) permits to define thg so-called slmpllctal symplgctic structoro on the

lattlco quantum phaso spacq (LQPS) in analogy with the ordinary symplectic

structure on tha CPS. It also reduces to It at the classical limit. As for equ.

(70), It gives Its continuous analog that permits tD define the true quantum

symplectlc structire on the quantum mechanical Hilbert space (QPS).

Sixthly, the Softwlngar basis may also bs considered as a Faurlar basis

and then becomes fundamental for thg Weyl-Wigner map :

R2d[71],

where Dp( a , a ) Is ths Wlgner density associated to ths symbol F( q , p )

of the operator CM q , p ).

This shows that quantization Is deeply tied to Fourier analysis, since

any operator is given as Fourier Bxpanslon.

Notice that the phaso ara in (69-7D1 plays the role of a quantum correc-

tion which is BxprBSSsd In terms of ths classical Poisson bracket [see £9}) :

oat C a*, b*) ; (cf, a*1) ] = -^ [ ^p*- &q> ctp*- ifcfjp (72).

Here, ths Functions F( q , p ) and G[ q , p ) under consideration are the most

simple linear functions on QPS.

On the other hand, [69), (70-72) reduce respectively to :

T{ a*, $) = exp{ - toa[ [ttQ)ii*,t)] T{ n, 6*) . T[ s \ o ) (73),

tq}p (74).

Generalizing (45] or (51.3) for d degress of freedom, any element A in

operator algebra is defined by :

A = - 4 5 3 ^ T . (75.1),

with,

Tr[ T J = tiP 6 , Tr[ A ] = a11* , a^*= Tr[ T ^ .A ] £75.2).

For any apgrator A, ana can define Its continuous analog by taking the

following continuous limits :

^ ft ( 7 6 ) sN

(77).r f^* N —• cc [2ir)d jR2d

Then, the operator expansion (75.1) appgars as the discrete -/Brsion of

the operator Fourier transform (71) :

ilm A=OP(ph, ?] = — ^ r f dtd^i at $) T( a", 5*] (78).N ^ 2 d JR2d h

It follows that the coefficients a^ , that are ftnctions defined an LQPS,

becomB at the continuous limit a Wigner density Dp( a*, 6*} associated to

soma symbol F( q , p ).

Then, If we want to deflng coinplatsly the quantum symploctlc structure

on OPS, wa mist firstly gsnerailzg the Formula (52] by taking the product of

two arbitrary operators A and B with coefficients a^ and b^ respectively and

then considering the continuous limit.

We may also use a mors compact notation for the above expressions In

analogy with that LEsd in f . 2 . Let e = ( q, p ) be a se t of local coordinates on

the CPS [see (2)), ?= lq>p) i t s analog on the QPS, 5*= ( a \ &) and

- ( c \ d \ Then, for Instance [67.6), (69) [70-72) and (71) read as :

=Gxp[t£x6*] (79.1),

T£ f) • T( a*) = exp[ Ia2[ ^\ / ] T [ ^ + / ) (79.2),

R2dC79.4).

\ the product of two operators Is given In the discrete and tha conti-

nuous versions respectively as fallows :

_ 1

5*1 .Dr.(75^ T( a*l.f

(2jr]T[ v) (81)

Then, the discrete Wlgner density cP and its continuous analog Drrt

are given respectively by :

/ 4 : V : * < ' I < ?:.?*)] (B2),

) , n rrt [2TT)U J

In general, a twisted convoiution prodrci" x l( is deflnsd by [see (22)) :

f d ^ F [ ^ ) . g [ " ^ - ^ ) exp[ i ^ A " a } (B4J.J L

Then, In [83) we are \n presence of a twisted convolution product with

a deformation paramoter v = f\ characterizing a "quantlzatljri' such that the

classical situation Is recovered when h • G.

As for [82), It describes also a twisted convolution product with s defur-

motion paramotBr v = £ o - expressing a "discretization" such that the conti-

nuous analog is raccvgrad when M • os.

Using [25), ona may deduce from the twisted convolution product x

between WlgTer densities a star-prodjct * betwaBn symbols.

Knowing that t r 1 [ ' ? ; o^j In (B3) Is given by [79.3), then the resul-

ting star-product *ft Is defined by (see (25}) ;

F(7*} ' ,G(Ti =FCt}.GCt] +l-^[FCt) ,GCt\ }p+.... {851.

The resulting algebra equipped with the star-product x^ becomes ncn-

cammutaitve. Assumed to be stable under "- , this algabra will concern only

Functions having compactly supported Fourier transforms such that this star-

product may also bs definad as the converging expression :

F\Glt)= X ? ? ? ?J d?d? F( ? ) G[ ?(86},

Is easily deduclbl© from (25) and using the Dirac delta :

(87).

Before defining the star-product **,, we need to define the "symbol", say

F( flt of an aparatar A (Ws will denote it AP In analogy with Op). In other

terms, ws need tD define the discrete version of [see (17)) :

^ D r r i ^ ) exp[ I i*x^*] = F[ Dp] [t) (SB).

The problem here Is that Fourier transforms are expansions In unitary

irreducible representations and in our casa, ws aro in presence oF protective

representations. Recall that, In order to have a Wsyi-Schwinger realization

of the Helsenberg group, one needs to perform an extension of the sbalian

double cyclic group ZwSZj^, so that the truly unitary representations would

be actually related to ZWSZ^t and not tD the Heisenberg group.

Knowing that the discrete Wignsr densities always convaiutg In a twisted

way with a deformation parameter v - -^- (see (82)), we may use now

a unitary representation and dBFlne tha discrete version of (88) by :

(B9J,

•where f*= ( ? , £} 6 Z ^ Z ^ , ?= [ rit ra,...,rd }, £= I s ,s s ), p*=

( nt, n'} (see (66.2)), <*> given by (34) and fCf*) may be viewed as a function

on ( the Fourier dual of) LQPS, [27].

Ths Inverse of the dlscrotQ Fourier transform (89) Is thsn defined by :

^^rVn/fl (90).

Moreover, If A* is an hermitlan operator, then :

and the symbol f Is reai ;

fktt)=nt) (92).

in fact, the relation (see (75.1)] :

^ a ^ T ^ = F [ 3 ] (93),

represents the discrete version of tha operator Fourlor transform [79,4),

so that one has [see equ.'s (18)) :

Af = F [ r 4 [ F J ] = F [ a ] (94.1),

F = F [ a ] = F [ F * [ A f ] ] (94.2),

and from (80,32), we recognize :

C^ = Af D Bg = F[ F*[ h ] ] = F{ r \ f ] x N r \ g ) ] (94.3),

h = F1hle = F i r 1 [ f ] « N r 1 [ g ] ] 04.4).

BefDra treating an example af this discretization by dofcrmabtan for the

case N = 2> let us study the commutators of operators from which one may

deduce the deformed Polsson bracket {,} . TbB commutators :

generalize respectively to :

(27r)

T{ ?*) , T£ S*l ] = 2t sln[ ir,[ ^ ; ^ ] ) T [ ^ + / ) (55.2),

195.3),

Tit) (95.4),

2 a1" ^ " w sin{ ^ [ / i ^ } [95.5),

X - ^ f d/rttC/) Dpt"?-^ sln{ a z[^; ^ ] } [95.6),

ropresent now the Wigner donsldes corrBspondlng to the discrete and

continuous versions raspsctlvely of the Moyai bracket {,} , which Is defined

in genera] by:

<f»e>v = i l r [ f i y e - e V ] t96}-In the continuous case, the deformed bracket Is not siss than the Moyal

bracket defined in f 2. [see (28-29)) :

[97.1),

with bhe classlc3l l imi t :

lim [ FCs) , G(t) }fi = ( F(l*] , Git) }p [97.2).fl •O

The Moysl bracket (97.1) may also be defined by tho. following convar-

glng relation ( see (86) ) :

te1 de" F( e' ) G[ ^ ) sin[ ^ ( c x ^ ' + ^ 'xf ' + c ' u ) ]

(97.3).

Using (82) and (09), ana can define the discrete versions af (36) and

(87) as follows :

v r (98.D,

Let us remark that (98.2) is a generalization of (61.3). Now, one may

derlvs the dgformed bracket {, }», slthsr using (95.5) or directly from (96)

and (98.1) In analogy with (97,3) ;

N f t N ^ ' (98.3)/

Finally, let us treat the cas© N = 2. Hers, we are dealing with F'auli

matrlc5QS a, = T10 , az = TM and tr, = T H { see (6Q) ) with one degree of

[ d = i ] . Using (42.2), one may easily SBQ that (see (52}] :

reproduce exactly tha su[2) algebra :

f a, , ffj ] = 2L Q% , [ ffj , a, ] = 21 aA , [ <7, , ^ ] = 21 <r, (100),

for rr.i, m a nt and n, deflnsd mcduio 2.

ThBn, it results from (45) that the coefficients a m n [Wlgner densities )

ccrrespondlng to crL) a2 and a, are respectively :

, n l (101.3).

The associated symbols are given by (see (89)) :

£102.1),

, (102.3).

Using (82), the different twisted convolutions are :

2 as = 2 i 1 2 <Sjl S - l l r J / 2 , =2 « 2 a, = 2 ^ d H

St e ( k - J > / 2 H03.1),

2 i 5 1 <SJ2 S

l r t - 1 | / 2 , S, ^ ^ = 2 ^ 1 i12 gW^2 (103.2),

(103.3),

Using (94.4), (99) and (42.2), the different star-products are :

(104.3).

Finally, the deftDrmsd brackets

fF I\ M » I

{f» , f

corns as

• 2 = f

• l - 2

• } 2 - T

1 S2 K

(sas [9

f,

•U

•ft

6) or (98.3j) :

(105

(105

(105

.1),

.2),

.3).

It Is sasy to SBQ that the bracket {,}-, obeys the Jacobi Identity (30).

WQ remark here that the symbols redefined as F. = iwL will verify the

sam9 su[2) algebra with the deformed brsckat {f}7 as do thalr associated

opcratDrs <7» with thB commutator [,].

This simple sxampla sho- ws how the Schwingar basis [ T } descrlbir^;

a quantum lattlae two-torus may be viewed as a set of generators of some

non-enrnmutativa associative algebra equipped with a star-product and defined

on the ordinary lattice two-torus.

In the general case, tha relation (89) represents a Fourier expansion in

the basis {<J* x * = exp[ i^f ju*x"?l } of functions on the lattice torus. From

(75.1), one may associate to a given Schwinger generator T^the Wigner

/i*= C m, n*), F*= (lc*»1*), m*= tmj,...,md}, rf= (n ,...,nd],Tc*"= (k«,...,kd)

and7*= [I ,...,td). The associated symbol Is then gl^en by fCt) = J* * XIt

Is remarkable that tha Wigner functions {symbols) corresponding to the basic

Schwlnger operators ara Just the Fourier basic functions. Moreover, to the

rszsnrwi S5 as w

product T^T^of two gsnrators of ths Schwinggr basis will corrsspond theu *

Wigner density c^ = N" & 'A " U ei3f=t p ' -U J and, If we dsnota ths symfaals

associated to T^and T^by fu and fu rospectivaly, It follows tha twisted

products :

and the deformBd brackets :

F and f

F [ n = SlQf3[ ; Ifijlf+fi*? (106),

FurthBrmore, one may deduce from [69) or (73) the following relation :

T[ 5 \ 3i.T£ a*, *) = q( ^ ' ^ x ( ? ' ^ T{ ? , B .TC £ , di

where,

q (109),

such that q • i when h • 0. Choosing : [28]

t=~5*=u,£=c = t (110.1),

wher u Is some unit vector, v/a gat :

T( a, if ).T[ u t o) = q TC if, o*).T( ? , u ) (110.2),

which coincides with the defining relation for tha Manin plane, [29].

In thg discrete case, the formula analog to (iiO.2) is not else than the

basic relation (32), with w = a 1 2 ^ — • i when N • « . Finally, It is

important to point out that the LQPS is a space with a matrix structuro and

this leads us to use the formalism of matrix differential geometry, [14-15].

5. Matrix formulation of s quantum bnsnnlc oscillator :

Let,

H = 2m~{p3 + m y q3} (HI),

be the HamiJtonlan operator of a 1-dlmanslonal harmonic oscillator, vAysre

the coordinate and momentum operators obey the fundamental Heisenberg

algebra [see (15)} :

The classical dynamical variables q and p tD which correspond q and p

describe a system with a mass m and an oscillation frequency w > 0.

Consider the change :

q = U U Q , p = (mfiw)* P (113).

Then, [111) and (112) reduce respectively to :

H = - ^ ( Q I + P2) (114),

[ Q , Q ] = [ P , P ] = 0 , [ Q , P ] = i * L (115).

Introduce now the usual craatlan and annihilation operators a and a as :

a = i ( Q . L P } (u s ) .(21* (2):

Then, [i 14) and (115) are rewritten as :

H = -3-r- [ a a+ + a+ a ) = to I a+ a + 1 4 ) (117),

+ + +[ a , a j - [ a , a j - u , 1 a , a j - il U L O ; .

Than, the sets { q, p, 4 }, { Q, P, fl } and ( a, a , 4 } describe

equivalent!)' bho Lis aigabrs of tha HBisenberg-Weyl Lie group.

In VIB'W of the commutation relations (1 IB], ths usual scheme for

generating ths quantum states af the harmonic oscillator Is based on the

properties of ths hermltian I occupation ) number operator v :

* = a+a (119).

These properties are:

[ a , i / ] = a (120.1),

[ a + , j / ] = - a + (120.2),

H = ftw(i/ + i f l ) [120.3).

Define the vaccum statB I Q > by :

a I 0 > = Q [121.1),

and ths excitad states I k > by :

(a }k

| k > = 3-1 0> , k = 0,l,...,N-l. (121.2).tk l )*

We have also the following identities :

| 0 > = jr 1 k > (122.1),

lit / I \ 4 i T J V f f T T T\

a+ ! k > = { f c + l ) * l k + 1 > (122.3) ,

f a ) n t k > = 1 i k - n > = * ( a ) n = 5 i k - n > < k i[(k-n]lj k"3l ttk-n)]J

(a !n I k> =[k+n)J

k!k+n> • )

n =

N-i fk+n)J"

(a } m U ) n i k > =

fc^O ^ k l J

[ kl ( k - n + rn )1 ] *

S k+n >< k I

[122.51,

( k - n ) lk - n + m> [122.6).

The quantum states I k > are elgonstates of tha number operator v :

J k > = k I k > (123) .

After this brlaf review on the 1 -dimensional harmonic oscl]]ator, 1st us

describe It now In tha context of a matrix algabra Mi,(C). We will use the

tlJde for the matrix expressions of the above-introduced operator quantities.

FlrstJy, the vacuum stata and the excited states are represented In this

context by Si-dimensional vectors :

a =

rcra

A

, no» =0 1

0m

,6

N » = 110 »

(124).

The NxN matrix representation of the operators a, a , Q, P and v are :

a =

(a i * o a oo a 21 a oo a o 3* o

1 {N-i

b ..o

a =

'0 D 0.1* 0 0.0 2* 0.

.0.0.0

D... ...".(N-i)* Oj

(125.1),

[Q i * 0...it* 0 2*

a 1I

u - —j[2)*

Q ' .

• . 0

D 0 (N-l j* Q- l j *(N-D-

, P = - L r[2)*

•'o

fQ -4* 01* 0 -2

o 2* G

0

.N-l

Q

o

(0 ...'(N-D* 0J

(125.2)

* P (125.3),

(125.4).

The commutation relations (112), (115) and [118) read In this matrix

35

where,

[ q , p ] = l f i H

[ Q , P ] = l «

[a i a 1 =

9= 11 =L(N-l)x(N-l)

0 0 i-Ni

(126.1),

(126.2),

(126.3),

(127).

Then, the matrix representation of the Hamiltonian DpBrator is glvon by

H = ^ [ a a 1a a ] =

M O O D ]| 0 2 0

0 0 3

2N-3 0.. 0 N-i

(128).

Let us remark here that the spectrum of the Hamlltanlan matrix H

presents a curious aspect due to the appearance of the matrix o In the

commutation relations (126). It comas that the eigenvalues af H are now

given by :

H f c k "

f2k+i

(129).

if k = N-i

k=2p-2

This Implies the following two possibilities for the energy IBVQIS :

N = 2p + 1

k=2p-l ^ - f i u

(2p+lJ

k=p-lk=2p-i

-fcv

lk=0 4

For Irstance, for N = 2 wg are In presence of ana energy lavsl with

a twofold degeneracy. 3n the case M = 3, we hava two energy levels - ftu and

2TJ- hu togothsr with a third one which is sn intermediate energy stats ho.

In ths Q-raprassntation, the quantum states ars represented by the N-

dirnGnsional vector :

= < 0 I k >

and with the Following normalization condition

<MQ> = 1

Then, the relation :

reads In terms of components as

Q

Comparing (133) with tho recurrence relation :

2 x«k(x) = 2

verified by ths Hermite polynomials H.M 'vvhich aro defined fay :

f^W = [-4)* e*

£130)

£131.1),

[131.2),

U31.3).

(132),

(133).

(134),

(135),

It follows that the states ^ i Q ) may be expressed ss :

fyQ)T 7 7 ^ r

Furthermore, It Is easy to generalize this treatment for a d-dlmensional

quantum harmonic oscillator. Then, for any l,j = l,...,d, one has :

[ q 1 , p j ] = 1 ^ 11 (137.1),

^ ^ 11 (137.2),

[ai,i*] = 5lj fl (137.3),

H = [ + mV f } =

(139),

[ ax , j / i ] = dj j a i = ax , [ a^ , v ] = - ^ a = a^ (140).

We define the vacxjum state and the excited statas by :

(141.1),

:=i

such that :

di kt k^> = © | k, > (141.2),

•.tllfH*"'" :»l i ^ - ^ " ••!* • » * • » -

aT ! 0.... 0> = Q forall t = i ,...,d (142.1),

10 0> (142.2),

where k. = 0,i,...,N.-i, For any 1 = l,....,d. Tho numbers N. may be viewed

as the components DF a prime decomposition of N (see (62)]. Then, d = i Iff

N is a prime number.

Ore has alsc the Following relations :

aj I k,...kr..kd > = (ty* I k4...tkj-l)...kd > (143.1),

j *"" j " ' d - j *"" j ""• d *

! Jq...(k1-nj...kd> (143.3),

I k4...0tj+mJ...kd> (143.4},

.kd> (143.5),

....k. > (143.6).d

k d ) •" ! ^ r'i^i *•

In the matrix scheme, ever/ sub-stats I k. > can b© represented by a N.-

dlmenslonal vector II k, >> with zero everyM^ere axcspt in the Ck+i)

position whsra DRB has 1. Theny the vacuum state {ID .D>> Is a tsnsorlal

sum of d vectors gach one representing a vacuum sub-stata with 1 tn the

first position and zero alse^'here and the I vector being NL-dimensional,

I = l,...,d. Ths matrix representations of, say, s , a, , Qj, P^ 2nd ^ are

glvgn by exactly the same form as in (i25.i)-(125.4) hut with N t In the

pisca of N, For H and v defined rapectlvely by {138} and [139], ore Is in

prosencG of a sum of d matrices, the I one being a N.xN- matrix and having

the same Farm ag in (128) and [125.4) respectively, where N is replaced by

N,. Finally, in ths Q-reprasentation, ths quantum stats-vectors OJ)

are roprasgntsd by a tensorial sum of d vectors, the i one $J.Qj) being

a Nj-dirnensional -*«ctor with components <& (QJ, L = 0,...tN.-l,defined by:

) = 1 £144).

In fact, the rQlations :

d J,£QJ = 0. icO.) , t = 4,...,d (145),X i' *• A X i-

give :

Q I *kI

and then,

-X72 72 ^ ( k l j *(kxlj

We conclude this section by remarking that ths GPS, equlvalently

described by the pairs [ q , p ), ( Q , P ) or ( a , a ), Is a space with

a matrix structure. Its treatment needs then ths use of the apparatus of the

matrix differential geometry, [14,15].

IWF5SF

6. Kfetrix dlfforontlal goomatry :

Per a general review on the subject, rsfsr to [30]. Hsreunder, one

presents a brief summary of this formalism.

Lot Mjg( C } be ths algebra End( C ) of all endomorphlsms of C^, i.e. the

sat of complex SIxN matrices, N i 2, ivU C ) Is then an associative non-

commutatlvB C*-algebra with unit f. Let { }, k € I = { i,2,...,N3-i } be

a basis of self-adjoint trscaless NxN mtrlces. Then, { 41, E,} Is a convenient

basis of Mj.( C ) consisting af hermitian matrices. One has the following

multiplication Sable :

+ ( 3 k lm - ^ C k l

m S E m (148),

Kr, are ths cx>mponents of the Killing form of suCN) glvon by :

Ky^j^^-Trl^Ej] (149).

In the cass whara K j = d^j, this basis becomes orthonormal and the

quantities S^m = S ] km and Ck]

m = - C ]km b&comB raai numbers.] k and Ck] = - C]k

The cagfflclants Coim GDrraspond canonlcaUy to structure constants of

su(K], i.e. :

[ , lEj ] = C Hm 0Em) (150).

Here, •*•& don't take interest in the notion Df the manifold itself , but only

in the algebra of functions. Actually, our functions ara not elss than

the matrices U» E^ and CMinear combinations,

Lst Der[ M^t C ) 3 ba tha aigsbrs of all derivations of W^( C ) In itself:

N(D) = ft £ EndfM^fD / xiE.F > = xE).F + E-xiF), 1> £,F E M^C)}.

DBT[MNCQ) Is then a Ue sub-algebra of the U s algebra EndfM^O of all

QfidomDrphlsms of M jCO such that ths U s bracket of derivations corres-

ponds to the commutator of linear operators. Since all the derivations of

are inner, It follows that the complex f resp. real ) Ue algobra

iCO) [ rssp. DerRtMN{C3] ) reduces to the U Q algebra sl(N) ( rosp.

su(K] ). The basis { a^ } of aJl darlvatlons aFM^tQ is formed by ths adjoint

action of ths ganBrstors { E^ }, k 6 I, of su(N) sDsrR(MN(Q). Then, there

are on]y N - 1 independent basic derivations Br defined by :n

Qk = ad( iE k ] (151),

and obaying the commutation relations :

In fact, the basic dorlvations s^ are in carrespandgnca i to 1 with the

generators E^ of M^(Q such that for any basic matrix Er one associatss

a basic d&riv^tlcn deflnsd by (15i) and with (see (150)) :

Qk( ]Et) = [ ^ k , iEj ] = C k j m ( l £ m ) (153),

and conversely, every basic derivation s^ af MM(C) Is of the form given by

(151) for some Er € M M ( Q . This correspondence is well Illustrated through

the simple example of classical Hamiltonian mechanics ••where every

classies! obserYrtbia F[ £ ) = Ff q , p} defines a Hamlllonlan vector flald Xp

(sea (6)) such that Xp[G) = { F , G Mses (9)), X p being fcho Infinitesimal

generator of the canonical transformation whose generating function is F.

Then, any element y °f DerntMtrtQ) will be written as :

and contrary to the commutative case, Der(Mvj(Q] do not form a

, I.e. a derivation muJtipliod by a matrix Is not a derivation.

Furthermore, It Is shown that ths smallest differential sub-algebra

r(Mtg(C)) of the complex C( D B T I M ^ D ) ; WN(Q ) which contains

Is the complex itself, l.s. :

N N D s r N [155.1),P ' P

^CD) =C°{ DarfN^CQ ; N^tQ ) E M ^ C ) (155.2),

= C( sltnjQ ; MN(Q ) = A B I W * ® MN(Q (155.3).

Any element a of Q T ^ ^ ( M M [ Q ) is a p-Jlnoar sntisymQtric mapping

f

and Its differential da € QDBrp+t[MfsjCCj] is defined by :

<* C be , x ]» xo.—»- *.p r s

X°'"-'Xp C DerCM^^Q) and " ?" meaning omission of x k , such that :

d=Q (157.2),

dE( x > = x< E ] = ad( IF] [ E ] = 1 [ F , E ] (157.3],

for any E 6 MNfD and x = adttF) 6 D o r t t ^ O ) with F £ M^IQ [see (151)).

In fact, each elamant a of Q^IM^EC)) is a finite sum of elements of

tha form :

a = So da, « daa - dafc (158),

with a^ € MN(Q and k € N.

Moreover, It is aiso shown that there is an operation of the Lie algebra

In the graded differential algBbra QQ^CM^fQ) In the seise of H.

Cartan [31], such that tha Induced operators 1 and L are matrix analogs of

the ordinary inner product and Lie derivative Df differential forms by

a vactor field % respectively. Then, we have the following relations :

(159.1),

(159.2),

(isg.3),

L £E ) = * [ £ » , for any E 6 MjgtC) and x ^ DertMfjtQ) (159.4),A.

or Is caiiod invariant If L £aj = D , for x £ Der(MN(Q) [159.5),

lv t . +ly V, =D (159.6),

/ i X* X J X-I L X-I * Xa J

Ly L -L L =L r , (159.8).Xi A3 Xs < i L X i » X J J

Now, In ardsr to construct the whole gradsd vgctor space

of mairix forms we use the differential d as define by the relations (157)

and the axterlor product. Firstly, define tha space QgBr [M j{C)) of matrix

i-Forms. Let { 6k }, k € I = { 1,2,...,N -1 }, ba a basis of 1-Forms Ami to

the basis { Qm }, m £ J, of real derivations (SSG USD), l.s. :

4

By definition, Qn™ (Hs]EC)) Is 3 mcxivie over MyCC), I.e. ona may also

doflne the forms :

E m e k = e k E m (161),

such that (saa (150), USD and (1S7.3)) :

dE k ( S j l= a j (E k )= l [E r E k ]=cy"E m (162),which masns that (ses (150)) :

This relation can bs invartgd to ylald :

ek = - - V KP9 Kfcr En.En dEn (164).N P r q

Now, the Grassmannian structure an QDer(Wn(C)) Is introduced as usual

by daflning the exterior product on the basis { 9 } :

ek * em = - em ft e

k dss) .

To guarantee the niJpotency (157.2) DF the differential d, one defines :

d( 9k * 9 m ) = d6k J m - 9 k , d0m £166).

Let us remark nora that, In general, we could havs chosen as a basis of

i-Forms ths set { dEfc }, k € I, but this letter prBsants a default due to the

folJowing non-commutativity properties :

4 WEJ ^ (167.1),

r,

when the basis { © } possesses the good propertlas (161) and (165).

If a is a 1-form, its exterior derivative da Is defined by (see (157.1)):

dar( «x » Oj 1 = et( n( e 1 ) ) - Sj( a( ©: ) ] - a[ [ e^ , S j ] } (168).

Using tho relation above, one obtains tho important identity:

k8P.©3PH

which is the analog of the Maurar-Cartan identity on the group manifolds.

Ths relations ii4B], (161), (163), (165) and (159) give s presentatton

of QQer[MNCC)) associated to the basis { E^ } .

The element Q DP Q^CM^JO) defined by :

B = Ek Sk (170),

is independent DF ths nho\cs of ths B . In fact, cne Pss :

0 [ a d £ l E ) ) = E - - ^ - T r [ E H l (171),

with E £ Mt-[C). FurthGrmDrg, & Is Invariant and any invariant element of

QQ [MfctEQ) is a scalar multiple of 9, This 1-form Is cailsd tha

Invariant alamort of QD g r (M^fQ). Using it, (163) and [169) raad :

eM N (Q (172.1),

dC-16) + E-16) 3 = O (172.2),

rsspectlveJy.

Finally, one may define an antillnoar tnvofutJon on DsrEMvrtQl and its

QXtansion on Qgar,[M^fC)) by setting :

X*( E ) = ( X< E#) )* (173),

ffp I X4 Xp ) = ( a p ( Xi 5 ^ p ])

respectively, with x, Xt^ Derff^tO), E € ^ ( Q and ap €

Then, QQ^CM^CQ} becomes a graded dfffermUal C -algebrs, with :

(175),

for <j € i rv (Mj^fQ), a E Qru.. tMtitC)) and 8 €

It results that a derivation x and a form or are called real if they obey :

X = X * U77),

ar = <i* (178),

respectively. Furthermore, if a is rBal thon dor Is also real.

Many more things which we shall not need here may be Introduced on

matrix spaces, such as : Integration of p-forms, canonical Rismarmlan struc-

ture for Mj^[Q, Hodge-star operator, coderlvatlvg, Laplace-Beitrsmi

operator, Hodge-Oe Rham decompasltlon, connections and their curvatures....

For Instance, the Integral of an (N -i]-Form may be defined using the

traoB :

r . » M 1 !E 0 «6 BiS 1 =Tr(E) (179).

We conclude this section by presenting very succlntiy the example of

N = 2, whsrg the basis of M3[C) Is formed by the 2x2 unit matrix H. and the

hermltlan traceless 2x2 PauJl matrices c , , j =1,2,3, tsee [60)).

In this case, one has :

a i f

(181.3),

(182),

(184),

8k (185),

= 2 € k lm o_ e r (186),

dsk = ^ e 5 , e m (187).

The relations [180), (iB4]-(iB7] give a presentoilon of

associated to the basis { u, }.

•60

7 . Quantum [matrix] sympiectic formalism :

Lst us discuss new the dlffsrant approachss existing m ths literature

scout the description of a qjartum symploctlc Formalism.

7.1. First approach :

introducing the canonical i-form 9 glvan by (170), Dubols-Vlolstte & ai

have shown that the 2-form Q :

IT in K. m nn K.

provldas a natural {quartuml symploctlc structurB an the matrix space,

dgflnJng consequently a matrix, (quantum) Palssan bracket.

Effectively, to sacfi element E t M^iQ will correspond a

vector field xE £ DerfM jtCS) such that:

all x

!n adriitlcn to this property, 0 Is closed by construction and then It

verifies the conditions stating the definition of a nan-commutattvo symploctlc

structure for M^jO g lvan ty Dubols-Vioiette [IS], Its expression given in

f. 1, is cbtained bu tsing a property of the star-prcduct [32] [see also [30]).

In analogy with the classical case (see f.2.)y ths matrix Poisson bracket

is then defined by f.vve use "m" for matrix) :

XF i = XE(F1 = - xF<E tl90).

61

Redefining a basic derivation a^ and any sloment Xp of D©r(M»t(Q) by :

^ (191.1),

XE = acK-jr-E) (191.2),

'we get,

[ E , F }m = Qfedt^F) , ad(£ El) = - ad(£ F) IE) = £ [ E , F ] (192).

This shows the direct correspondence between the (matrix) operator

algebra ©quipped with the commutator, and the associated (matrix) algebra of

symbols equipped with the matrix Polsson bracket. Hence, tha natural sym-

plactic structure for a matrix manifold is Just the commutator.

FinalJy, remark that the quantum sympleetlc mairlx

Is now a IN -i) x (N -1) hypermatrlx whose gntrles are the NxN matrices

up to a scalar which is tha value af the oorrespondlng structure constant

r k

in generai, the hiypsrmstrix 11 has an inverse defined as a hypermatrix

Amn with the propertlas :

"P f P (194).

'J'slng the Killing metric and Its Inverse :

' m n - " n m « — n (195. 1),

get,

£n

.En - Kmr K"5 Er.Es - Kmn 41 + i SmnP - . ^ c 1 7 1 ^ J E {196.2),

Cmnq = CmnP Kpq

Kpq ( totai l>' symm9trlc) (196.5],

and consequently, the inverse of the matrix E is jist Em, I.e. :

Then, the h>permatrix Amn satisfying aq. U94> is defined as follows

1N

and,

,2mn = Kmr ^ns ^ = N»( .nm _

Usually the inverse of an antisymmetric matrix Is another antisymme-

tric matrix. This Is not necessarily true for hypermatrlces. Indeed, althoqjh

the hypermatrlx U is antisymmetric, its inverse Am n Is no rnDra.

For Instance, for N = 2 we have

and,

mn

0

" " 2 gmn

, 0 cr4

3 - ^ 0

iT H n m

, A"1" =_ i-iff,

(198.2),

(198.3J,

(19B.4).

7.2. Socood approach:

This approach corslsts to Farmulats thg Wgyi-Schwingar reaJlzatlan of

the hslsenberg groqp [see section 4) In a matrix context and to adapt It to

the formalism of tha matrix dtfTsrentlai symplsctlc goometry presented In

section 6, [27].

The Schwingsr basis { T m n / m, n = 0,i,..MN-i } given by oq. 140)

may be considered as a basis of tha matrix algebra M| (Q consisting of

«htrmltlan matricss. Excluding the unit matrix Tw , the Tmn*s are N -1

inltary tracsless NxN matricss and may be viewed as as a basis for su{N).

Then, the relation (45) gives the expansion of any elament of M^[Q in

ths Schwlnger basis [ T fnn }.The multlpllcatjon table of these basic alsmgnts

&>**

Is given by t Q generalized -imposition law (52). Hers the numbers m,n are

••iefined modulo N. Krcm nyu, '-vs will use the compact notation given by

oq.'s (SO).

Lot us first define tha Lis aigsbra at* derivations of IYWC) relatively to

tha basis { T ^ / ~rr\ -f- o }. Define the basic derivations as given by eq.'sm

U5i> and (153), i.e. :

^ _ ) / B_^ tTJ= [ T _ ^ T J (199).m m m p m p

^ , T^ j = - 21 sln{ aa[ m , n ) } T ^ (200),m n m+n

and its Jacob! Identity*

m n p

one aasily Find that the basic derivations e^obsy the following relation :m

[ s_^. a^J = -21 sln{ az{ nf, r?) } e_^ ^ i202).m n m+n

Comparing with eq. [152), WG get the following structure constants :

C_J> = -21 sin[ cr3( m% n) } 6^ £ (203),

mn m+n

\vhere the Kronecker symbol is defined by oq- 1.51.5).

We will nov.' compute the Killing metric for su(N) relatively to the basis

i T_+ / vt? *) by using :m"

Tr[ act A ;.ad£ fi ) ] = 2H Tr[ A.B ] = 2 2 K_^s m bn i2G4),

•where A, B t M|^[Q. it results from a direct computation that K is givenmn

by (see oq. (196)) :

mn 2N ^ > mr n s

Using sq. 1.149), i.s. :

- 1 T

mn m n

ths relation [200} and the property £47.1), i.e. :

-£j- Tr[ T_^J = 5_^+ (207),m m,o

•we find that the Killing metric is given by :

mn

Remark that the metric g^,^ defined on the oporator algabra by eq. [45)

Is now tied to the Killing motrlc by the following relation :

mn m,-n mn

for m, n t o.

It reinsins to dstsrmlne the gensraJ sxprssslon of ths symmetric

f. Using the relation :mn

n n m mn mn p m+n

it follows that al] the quantities S_^_j; identically vanish :mn

J ^ _ ^ / (211),mn

slncB the indices m, n and p are defined such that the vanishing index o is

excluded.

We finally obtain tho following multiplication table :

+ V ff»i n , rS Jj T^ ^ - ^ ^ J ^m n m+n mn mn mn p

• ^ orai Hf, n) } 6^ ? T ^ (212).m+n,o m+n p

Let us introduce now a basis of a 1-Forms, denotsd by 8 m , dual to the

basis of darlvatlons { B_^} tsee the relation (159)). Of course, the indices mn

and ffare ai'*<ays deflngd such that the value o is Bxcluded. Than, we have:

S™ (g^J^^fl (213),n r,

• *

such that the 0 m satisfy the following relation :

where " « " is the extarior product Dn matrix forms.

SlncB ^g / tMf^ lQ) ^ a M^tO-modulB, one alsD has :

T_ | F S n =0 n T_ | ( (215).TTI TY1

»- M

67

ff sJso easy to sea that :

n in n m

sln{ sr3( n\ m ) } T^ ^ C_^J T + i2t€),m+n n m p

sc that ;

-'2 C+Jf+eP 1217).m jjy mn p

aq.'s (165), (168) and [203 j , ore* finds

d 0 ^ = - 4 5 J ^ ^ ^^ pq

= - 1 5 sln{ ffa{ m , q) } e^ . e m - q i218).

Thon, th9 oq.'s (212), {214], {215}, (217) and (218) give a

of O-, (Mk,t01 assoclatad to the Schwlrver basis { T / m"* o'}.

Tha Bl9ment 0 fc 1 2 , ^ 1 ^ ( 0 ) defined by :

m

be called ths canonical i-farm. As It is axpscted, this 1-Farm plays tte

rcio of the Llouvllla 1-form and Its differential, ths 2-form :

il = - 49 (220),

68

•will represent the qu&tum [matrix ) symplectlc 2-farm. It Is given by

mn mn p nrt

t n) } T_^ ^ 8 m ^ e n (221).m+n

2 2Then, the quantum sympiectic matrix Q_^_^ is a ( N - i ) x l N - i )

mn

symmetric hypBrmatrix: whose entries are thB NxN Schwingsr rnatriCBS :

(222).mn mn p n m n m nm

Using aq . ' s (222) and (51.3) , WB find ths following reJatlons :

3 + < A ) = - R " 2 n++*n ( 2 2 3 )*rrs + mn

[A,B] = 4 - 2 a m Q ^ b n (224),

= 5 Q 4 / (225),m -^ mn

n

(226).

mtn mri

The h^'permatrix Q^^has an Invgrs9 vi^ich Is defined as a hypermatrixmn

A m n satisfying ths following propertiGS :

** + + j ^ (227).

in crdsr to determine Lt, criB must first define ths inverses T111 and K

of the Schwlnger matrix T^ and tha Killing metric K^_^rsspectlvely.rn mn

The Killing niQtrie, which may bs used to raise or to lower Indices, has

an Invars© defined by :

W ^ * \ 122B. 1),np pn p

Q2B.2).

and tne in'/erse T01 Is defined such as :

m^ = K m n T 4 and T^ = K ^ T 0 (229),n m mn

;o that ths T m ' s ara just the hsrrrltian conjugate of T_^ (ses ©q. (41 j) :m

T ^-fJ'T^ (230.1)»-m rn m

4

t23fl.2).

Lot IE remark that the eq. (.61.3) may be rewrlttan in this context as :

2 9xp[la1tm,n'H =N*(5 -1 (231).

Using eq.*s (205] and (212), Dn9 verifies that the hypgrmatrix Amn that

sq. 1.227) must be of the form :

7O

7T ' N3" r"' S

_ 4 g lJ j i m , n ) Tm+n

4 - B*pi x i ffixn*]N N

N p*

- - V ( ^ * . ^ 4 . i sm { azi ^ , ^ j T ^ 4 ^ ] (232).N

Notice that this hypenmatrix Is not antlsymmetrlc.Indeed, ong has :

A™11 = axp[ 21 a,( if, in ) ] A m n (233).

The aq. (217) (or (225)) can be inverted to yleJd ISBB oq. (164)) :

m N rn

N1711* Kn s T_^. 1 \ dT^ (234).

Finally, If WB consider N = 2, with I*= (1,0), 2^= (1,4) and 1^= (0,1),

>ve attain the same results I^J to a factor " X " as givan in (198.4).

Furthermore, one may aiso vorlfy that thB quantum canonical i-farm &

defined by (219) Is effectively the basic Invariant ctona* of Q (

indged, for any ^/ector field x G DBrfM^tC)) glvBn by :

piP'liPpV^^I^V^Pr PH

7 i

m

-xr5 x Q en = 4 r5Y m C

and,

mn

e m

on© has,

i f t ^ (236.1),

X ^ / T ^ / (236.3),

so that, for any y G DsrtNVjtC)), OTS has :

L t e ) = [ l d + d l ] t d ) = O (237).

Moreover, It i s easy to see that:

LXL Q 1 = - L t dS j = - d I Ly[ 9 j ) = 0 (238),

and conssquontiy, all vector flstds y are said, In thg classical tsrmlnoJogy,

ID bs sirtctiy Hamiieonlan ,[33], ar globally HamlfCorHan, [ i j . This -means

that there Is an Isomorphism betwean i^rw (M^O) and Der[N4,(O), SD that

to aach vgctor x £ DertM^Qj will correspond an eJement a - dA of

%>.» tM^fQ), where A is an elamont of the operator algebra such that

oq. (239] is satisfied, and conversely, to each 1-form dA one associates

HamlJtonian vector field xA such that:

a = dA (239),

where the components x ^ m of x^ in thg basis { 9^ J oolnclds with the

coefficients am of A In the Schwinger basis.

Honce, operators of the form dly) play the role of generating functions :

•i[ «( Xi) K Xi * = Q< Xi . Xx) = - d[ 0( xi) ]( X. ) 1240),

fbranyxi, Xa^Dgrft^iO).

The reason v»hlcfi motivated tha choice glvsn by eq. ii99) rather than the

one given by eq. tl51} Is to ba In conformity with ths results given by eq.'s

195.31 and (95.5) for d-i , i.e. :

r" r m 1.241).

In analogy with tha ciasslcai case, the corresponding tjiKintum (matrix]

Potssan bracket is then defined by (see Bq. (.190)) :

{ A , B ) m =: U{ xB , XA ) = XA(B} = - xB(A) (242).

As expected, this matrix Potsscn brocket Is just the commutator :

S C = 4 - 2 ^ b™ «in[ a3i ? , r* ) «T (243),

and we can read the discrete version of the Moyai bracket campongrts ise©

eq.'s 1.95.3) and {95.5)).

Thus, ths quantum tmafirixj sympiectic 2-forrr, Q doflrad In the context

of ths non-commuCatlve dlfforartlai geometry of matrix algebras M^tC),

'which has fcsen Introduced by Oubals-VLoletta S al, givgs directly the

discrete version of the Kioyai bracket.

in other rBspocts, It is easy to verify that the Jacobl Identity :

[ A . I B . C U + c.p = 0 1244),

is equivalent to ths relation :

= - xc( { A , B } m !

i r j

r'cr aii cporators A, B and C. Jn particular, ',ve hav-a :

TTi

m n q q m n '"

= « ^ t 1 ^ , o l ) = - dt {T+ , T + L )t G J 1246).m n

W« dsdsjre frojv, gq. i;245} that the matrix Polsson bracket plays ths role

of trie generating function of the Lie bracket of ths corrBpcndlng HamlJtonian

'.•ectDr fields.—Me

QTD may also Introduce thQ antisymmetric hypsrmatrix i"2m' defined by :

•+-*• - + - • -*•+ •*• •* , - • • > - * - • - > - •

t i m n -= K m r S n s Q_== I Tn , f ] r N { A m n - A n m } = - Q n m (247).

In order tc carnpists the analogy with the oiasslna] case, ws givs hers-

urKteir the quantum analogs of the classical relations (6-7], (9) and [12-13J

respectlvgjy :

m

^ ' XA J (248),Vd

.ron (n

" XBi A ] = e ^ (A) A'1"1 e^B) [249),

LslbnltzA , B.C U ^ ( A , B }m . C + B . { A ( C | m

Cj [250).

the qiEtum analogs of sq.,s (5) and [10-1 i) sro already given by the

sq,,s 1.2210 and [244-245: respectively.

in other respBcts, it. Is straltghfarward to taxs the continuous Jlmlt.

Firstly, Dperste the cnango given by sq.,s I.65J and [66) for 6 = 1, snd

taka the limit N * » given by sq. ts i.67). Than, the continuous analog of

ths Sch"/!ng^r basis !s the set of the non-trivial translations operators

the quantum i c-mmutation rslatione raad :

I '"Iff" f-i 'Y<,-> . t 1 - -"It r-WJi. i' ",r1 _ *-o \ ) ^l 'a j .n.a t ^ i h j J - f l <1jTfa F* l" 5 t !

Thus, the quantum structura constants can Ge dsfinBd as :

C, ,y , ^ , . = -21 sln(4 tad - fac; j fla+c^.tffti-Ht-f) (252)..-•ajJi UjW i^jj; ii

Mcraover, the quantum Silling Form takes the form :

^ I i ^ t2S3).

"This metric is also prGportlored to the quantity :

do do' ds cb'C, bWrt w „ C, , fa,Un, , j r g> 1254),

In arajcfy with the rsiaticn it95.2).

Finally a quantum derivation sf_ ^ and a qLantum diffaraitlal form

3 L S V J ara defined as :

a, M - I Tla.bj , . j = Adi T(a,bi J '255),

8. Dlscjssion and pQrspoctlvBS :

The aiscrsts Weyi-fteisenbQrg algebra raay gi'/e same good understanding

•:f the nc-ticn cf the quantum phase space, [34], This latter is necessary to

ccmpJetsly formulate quantum mechanics.

Fbrtharrrcrs, \vs find actually In the iltarature many attempts tc

describe quantum mgcnanlos. The stsr-dsfcrmacion, quantum algebras and the

rcn-commutative differential gsomstr/ are essentially thg toots used for this

program, [15], [35], [36], [37]. It 3s also argued that it is possible to

describB classical and quantum mechanics in s unified scheme of nco-commu-

tacJve geometry, [38].

More generally, gangs theories are aisc the subject af same recent

applications of quantum grtx^s, [39].

The Newtonian and the Lagnanglan formulation of the quantum

mschanics in this context are also treated In some recant papers, [40], [41].

Finally, quantum mechanics has been aiso described in the context of

soma relatively new notion, that Is, braided groups and aigabras, [42].

We plan to treat all these directions in a second paper.

Acknowledaments

The author would like to thank Professor Abdus Salam, the

International Atomic Energy Agency and UNESCO for hospitality at

the International Centre for Theoretical Physics, Trieste, where

part of this work was elaborated.

.„• «j ij. > ^ !

77

Rafarancos :

[ i j V. Ckdllamln S 3. Stamberig : "Symplecttc techniqu9S In Physics",

Cambridge Univ. Press, Cambridge (1984),

R. Abraham £ J.E. Marsden : "Foundations of mechanics'', Addlscn-

Waslgy Publishing Company, 2nd edition, Redwood city.Callfomia, [1985),

V.L Amoid : "Mathematical methods of classics] mechanics", Sprlr^gr-

Vsrlag, N.-Y., [1989J;

[2] H. Weyl : "'Theory of groups and quantum mGchanles", Methuen, Lcndon,

11931);

[3J E.P. Wignar : F%s. Rsv. 40(1932)749,

See also :

G.A. BEker: Phys. Rav. 109(1953)2198,

G.S. Agarwal & E. Wolf : Fhys. RQV. D2{ 1970)2161,

0. GaiatU S A.F. Toledo Plza : Ffiysica A149(1988)267;

[4] J.E. Mayai: Prcc. Cambridge Phibs. Sa?. 45[1949)99,

See also :

D.C. Rivlor : Phys. Rsv. 83(1951)852,

T.F. Jordan S E.C.G. Suriarshan : Rev. Mod. Phys. 33[1961)515,

A.E. Ramisr : Am. Phys. 95 [1975)455,

K.C. Liu : J. Math. Phys. 17 [1976)859,

P. Sharan : Phys. Rav. D20[1979H14;

[5] A. FraaildTsr S A. NljarfxUs : Proc. Nat. Acad. Sd. 43(1957)239,

fC. Kodalra S D.C. Sparer : Am. Math. 6?[1958J32a,Ibld 71 (1960)43,

M. Ggrstorfefaer : Ann. Math. 7Bti963)267,Ibld 79(1964)59, Ibid 84

11966)1, Ibid 88(1968), Ibid 99(1974)257,

A. Nijanhms S R.W. Richardson : Bull. bmsr. Math. Sac. 72 [1966)1,

J. Math. Mech. 17(1967)89;

[6] J. Vey : Comment. MatK Helvetia 50(1974)421;

[7] M. Flato 8 al : J. Math. Phys. i7!;i976!i754 t

F. Bayan & a l : U t t . Math. Phys. 111977)521,

F. Bayan S si : Am. Phys. 111(1978)61-152,A. Llchnerowlcz : "Quantum mechanics and deformations of geometrical

dynamics': in " 'jiantum theory, groups, fisjds snd partioi9s",A.O. 9arut Ed.,

D. raldGl Publ. L'c, Dordrecht, i i9S3j;

[BJ J. Bafcas S a l : J. Phys. A2O[1987j3713,

G.V. fXm : J. Phys. A21(1998)2321,

Ci.V. Dura S a i : Phys. Lstt. 8211(1983)445,

U. Falrlis S ai : Phys. Lett. B224li9S9>101,

j . Bsfcss : Phys. Lstt. B228 {1989)57,

J.F. Carinara & a i : RalatlvlstJc quantum kinematics In the Moyal repre-

sontafcion" Zaragoza Univ. preprint DF7UZ-99/5,

0. Gurevlch & V. Rifctov : J. Geom. Phys. 9(1992)25,

A. Baiiesteros S ai : J. Math. Phys. 331.1992)3379,

J.-P. Kammsrar & M. Valton : "Representation d'uns partloila quantique

ds spin i par Is prcdui!: dg Mcyai st uns supsr-varistB sympiectlqus piats"

ic appear in J. Gecm. Phys. ;

[9J M.A. Rteffol : Caradlan J. Math. 4G[1988)257;

[10] M.A. Kleffai : Gommun. Math. Phys. 122(1989)531;

[11] A. Conres : C. R. Aoad. Scl. Paris, t290, serle A [1980)599 ; Publ.

Math. IHES, 62(1986)25? ; N'3somgtrle non-commutative", IntarEdltlons,

Paris 11990);

[12] D. FalrilQ S al : Phys. Lett, 6218(1989)203,

.1. Hopps : Jnt. J. Mod. Phys. A19H989J5235;

[13] E.G. FInratos S J. IllopouLaB : Phys. Lstt. B2D1 [1988)237,

J. Hoppe : Phys, Latt. B215(1988)706,

E.G. Floratos : Phys. Lstt. 8232 [1989)467,

[14j M. Dubois-Vioiattg : C. R. Acad. Scl. Paris, 1307, serie I (1988)403,

M. LJufaols-VlolottQ a ai : J. Math. Phys. 31(1990)316 ; J. Math. Ph>-s.

31 [1990)323 ; Phys. Lett. 621711989)485 ; Class. Quantum Grav.

611989! 1709,

[15] M. Dubols-Vloistto : "Non-cc-mmutative dlffarantlal geometry, quantum

mechanics and gsugs t-hsor/' In Prcc. en "Dlffsrantlai Gacmstrlc Methods In

ThooreUcai Physics", C. Bartoocl a ai Eds, Lecture Nates In Physics 375,

•Springer-Verisg, [i

life] M. Ujbcis-Vioigtte : Lett. Math. Rhys. 19(1990)121;[17] S.L. Woronowlcz : "Group structure on ren-commutatlve spaces" inProc. on " Fields and geometry", A. Jadozyk Ed., World Scientific,Singapore,11996) ; Pub]. RIMS, KyotD Univ. 2311987) 117 ; Commun. Math. Phys.111(1997)613;

1!8J F.A. Berazln : Sov, Phys. Usp. 23(1990)1991;[19] J.-P. Karamsrer : J. Math. Phys. 2 [1986j 529[20j W. Haisarberg : Zs. f. Phys. 33(1925)879,

M. Born S P. Jordan : 2s. f. Phys. 3411925)858,M. Bom S al : Zs. f. Phys. 35(1926)557;

121] P.A.M. Dlrac : Proc. Roy. Sec. A109C1926J642;[22] J. NUadsrla : "Quantization as mapping and as deformation" In Proc. en"GiBntum thscjy, groups, flaJds and partlcJes", A.O. Barut Ed., D. RsldalPuti. Co., Dordrecht, (1983);

[23] J. Schwir^gr : Prac. Nat. Acad. Scl. 46(1960)570 and 893; Ibid 47s 1961)1075; "CXjantum kinematics and dynamics11 .Benjamin, N.-Y-, (1970);[24] R. Jacklw : "Chsm-Slmcns and cccycles In Physics and Mathematics",M.I.T. prsprlnt [1996);[25] R. Aldrovandl & D. GalatU : J. Math. Phys. 31(1990)2937;[26] J. Patera S H. ZassQrihaus: J. Math. Phys. 29(1988)665;[27] R. Aidrovandl : "Quantum Hamiltonlan stricture" and "Quantum HamlJ-tcnlan dlffsrsntlal georngtry : How dees quantization affect phasB spac©'?"Estadual Paullsta Unlv.prQprlnts (1993);[28] A.E.F. Ujemal : Mod. Phys. Lett. A7(1992)3169;[29] Yu.1. hfenln : "Quantum groups and nDn-ccmmutatiV'O goomatry",McntrsaJ Univ. praprtnt CRM-1551(1988);

[30] A.E.F. Djamai : "Introduction to Dubols-Vlaiette's non-commutatl1^differential geometry" Oran Univ. preprint [19941;[31] H. Cartan : in "CoJlv-qus ds TcpoJcglB", Bruxeiias 1950, Masson, ParisU951);

[32] E. MDUTTB : Am. Jnst. Hgnrl Polncars 53 [1990)259;[33] A.A. Klrlllov : ''Elements of the theory of rsprGsentatlon", Splnger,Berlin, 11976);

ao

>34] B. HiJgy 3 N. Mcnk: Mod. Phys. LQU. A3S[i993j3625-3633;[35] N.P. Landsman : "Deformations of algebras of observablss and theclassical limit of quantum mechanics" , DANdTP-92-14;[36] M. FiatD & Z. Lu: Lett. Math. Phys. 21 {1991)85,

M. Fistc & D. Sterholmor : Lett. Math. Phys. 22 [1991) 155,

Z. Lu : I. Math. Phys. 331:1992)446;[37] A, Dlmakls 8 F. MuUsr-Hoissen : J. Phys, A 25[1992)5525;[38] T. Dass : "Unified formalism for classical and quantum mechanics In agsngraiizad sdnsms of ncncomrautative goomstr/', Ksnpur preprint (1994);[39] I. Ya. .Arefova S I.V. Volcvich : "'Quantum group gaugQ fields", SMIpreprints [19901,

L. Castellan! : "U tN) gaugQ theorigs", Torino preprint DPIT-74/92,T. Brzazinski S S. Majld : "Quantum graup gauge thgary on quantum

spaces", DAMTP/92-27,

L. CasteJlanl M.A.R. Montelro : "A not© on quantum structure constants"OFTT-18/93;

[40] A. Sudbsry : "Quantum differential calculus and Lie algebras", PRE-33667-York;

[4t] M. Lukin S ai : "Lagranglan snd Harnlltonian fomnallsm on a quantumpiano'1, Univ. Alabama preprint, UAHEP-931 {1993},

K.P. Mailk : Phys. Lott. B316ii993>257;[42] i;. Majid : 1 Georn. Phys. 13(1994)307,

5. Majid : "ijuantum and trsidod Ua aigofc-ras" 0AMTP/93-4.